Amine Beyhom S.urmavi s.151

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http://sas.academia.edu/RichardDumbrill/Books/182486/ICONEA_2008

A NEW HYPOTHESIS FOR

THE ELABORATION OF

HEPTATONIC SCALES

AND THEIR ORIGINSAmine Beyhom

...The double octave will not comprise, in practice, more than fourteen intervals; theoctave, more than seven; the fifth, more than four intervals and five degrees; the fourth, more

than three intervals and four notes; the tone, more than two intervals. It is experience and not the

theoretical need which dictates it [...]Ibn Sn (Avicenna) Kitb-a-sh-Shif [11thcentury]1

The reason for having eight notes in one octave is an arbitrary concept. There are

diverging explanations but none is satisfactory. The first part of this paper offers another view

based on the authors theory of Modal Systematics,2 where basic principles are explained. Thesecond part is a statistical analysis on the combination of intervals within the span of the just

fourth, fifth and of the octave.3 The conclusion proposes two hypotheses, the first on the

elaboration of the heptatonic scale and the second on the origins of heptatonism.

Introduction

The reasons given as to why the modern scale is made up of eight notes areunconvincing. Some suggest numerical relationships and their properties and others acoustic

resonance. There are also propositions stating the obvious: it is as it is because it cannot be

different.

The first reason is based on the properties of numbers. It offers two alternatives, firstly

the magical properties of numbers, and secondly the ratios between them. The first alternative is

dismissed because it does not relate to musical perception.4 Since Greek Antiquity, the

secondalternative has been the source of an ongoing dispute between the Pythagorean and the

Aristoxenian schools.

The tetrad which was used by the Pythagoreans and their European followers provides

the ratios of the predominant notes of the scale, as the Greeks perceived them (fig. 1: 176).5

However, it does not give any clues, and no other theory does, as to why the cycle of fifthsshould end after its seventh recurrence.

Later developments led to scales with twelve intervals, as in the modern European model,

and seventeen with the Arabian6 and Persian paradigms.

There are no reasons either for the fourth 7 to be made up of three, or for the fifth to be

made up of four intervals.

Then the Aristoxenian school raised a point of particular importance when it pointed out

that the practice of performance and the perception of intervals are the keys to theory. 8

The Pythagorean construction of intervals, which in part is based on superparticular

intervals,9 misled many theoreticians10 into believing that acoustic resonance might explain the

construction of the scale, on the basis of its similarities with it. However, this is inconsistent with

the predominance of the fourth in Greek theory and, for example, in Arabian theory and practicetoday. Acoustic resonance shows that the fourth is not the consequence of a direct process.11
http://sas.academia.edu/RichardDumbrill/Books/182486/ICONEA_2008http://sas.academia.edu/RichardDumbrill/Books/182486/ICONEA_2008http://sas.academia.edu/RichardDumbrill/Books/182486/ICONEA_2008


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To put it simply, Pythagorean intervals are based on a relationship of numbers based on

the tetrad, hence the following: 1:2:3:4. Therefore, any number can have a relationship with any

other in the series. Figure 2, page 176, gives an example of an extension to 5 consecutive

numbers: 1:2:3:4:5.

With time, new ratios appear. They come from combinations of the number 5 with the

original four numbers in the tetrad. However, whenever acoustic resonance is assimilated with a

generative theory, the only new ratios to appear are exclusively the consequence of number 5 inrelation to the fundamental, (fig. 3: 177), and even if multiples equal to the powers of two of the

frequency of the fundamental (upper octaves) can be equated with the fundamental, new ratios

can appear but always excluding the fourth12 (fig. 4: 177).

There are strong arguments in favour of the consonance with the just fourth. 13 However,

acoustic resonance fails in that neither can it generate modal scales, nor can it give satisfactory

answers as to the number of eight pitches in the octave, or three in a fourth.14

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Part I. Differenciation, combination, selectionand classification of intervals in scale systems:

basic modal systematic

The study of interval combination within a fourth or a fifth would have entertained

scholars since music and mathematics were found to suit each other. Aristoxenus had limited

combination techniques for his understanding of genera,15but Frb saw them as systematiccombinations.16 The combination of intervals must obey rules. Thus heptatonism is made up of a

small number of consecutive intervals which we shall call conceptual. They are placed in larger

containing intervals, such as the fourth, the fifth or the octave. Aristoxenus used the quater-tone

to define the size of conceptual intervals as well as for common denominator. With Cleonides itwas the twelfth of the tone which was his common denominator17 Frb divided the octave in144 equal parts18. This is twice the amount as in Cleonides. This shows how greatly Frb wasinfluenced by the Harmonists, as Aristoxenus had them labelled. These scholars were focused

on tonometry and generally used a small common denominator for a maximum of accuracy in

their quantification 19 of intervals. However, the Aristoxenian school favoured the largest

possible common denominator, i.e., an interval which can also be used as a conceptual interval (a

second among intervals building up to larger containing intervals such as the fourth, the fifth or

the octave).

Let us take a genus with a semi-tone or a quartertone as largest common denominator,

within a fourth. To find out how many semi-tones make up a fourth, add semi-tones, one after

the other until the fourth is filled up (tab. 1: 174). These intervals make a form of alphabet the

letters of which being multiples of semi-tones. In table 1, page 174, the intervals labelled 1, 2,

etc., are integers. They are multiples of the largest common denominator which is the semi-tone.

If we place three inter vals in a fourth, other intervals may not fit in any longer. For example, if

we place two of the smallest semi-tone intervals, the largest interval to fill up the fourth is one-

tone-and-a-half, that is three semi-tones. When a fourth is made up of three intervals, the

alphabet is reduced and has only intervals equating to one, two or three semi-tones.

The genera made from the systematic combination of the intervals in the alphabet

constitute the well known six genera of semi-tone scales (tab. 2: 174), among which the first20

and the fourth 21, are mentioned by Aristoxenus. The first three genera22 have two classes of

intervals: the semi-tone class, 1, and the one-and-a-half-tones class, 3. This also applies to thethree other diatonic genera, but in this case with intervals of one semi-tone, 1 and one-tone, 2.


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Interval classes can be expressed as capacity vectors, according to the number of intervals of

each size they have (tab. 3: 174).

Another approach to the problem would devise a literal expression for the size of

intervals expressed as multiples of the semi-tone, and then, arbitrarily, assigning the system

amounting to the least integer number, as indicator of capacity. A good example is the genera

with two one-semi-tone and one one-and-a-half-tones additional intervals (tab. 4: 174). The

digits of the intervals are concatenated in a single integer. The lowest number in the series ofthree is 113. If we assign the smallest number in the series as a capacity vector, we need only

count the number of occurrences of each interval. We start with the smallest one to find out what

is the capacity of the corresponding scale systems. This is known as a hyper-system.

Taking, for example, vectors 2,0,1 and 1,2,0, with corresponding hyper-systems 113 and

122 as basis for generating remaining combinations. The intervals in each hyper-system can be

combined differently in three subsystems, or unique arrangements of intervals contained in the

hyper-system. The reason for this is that each model contains a semi-tone which is repeated, in

the first hypersystem and two one-tone intervals for the second. The outcome of the combination

of intervals in a hyper-system containing three different intervals would be different. However,

this configuration does not exist for semi-tone integer multiples.

Conceptual, quantification, and

elementary intervals: Understanding theory

and practice

In the Western equal temperament scale,23 also known as the 12-ET system (equal-

temperament with 12 intervals in the octave), both conceptual and quantification intervals may

have the same value. The semi-tone is half of a tone. It is the smallest interval and therefore

divides the fourth into five semi-tones. The fifth, is made of seven semi-tones: three-tones and

one-half-tone. The octave has twelve semi-tones, that is six tones. The cent being equal to one

hundredth of a semi-tone, appears to be more accurate. However, it has little purpose with the12-ET since the semi-tone is the exact divider for all larger intervals.

With other systems24, the smallest interval used, in theory, may neither be a divider of

other intervals, nor a conceptual interval, or an interval which is used in the scales and melodies

of a particular type of music. An example of it is the systematic scale defined in the first half of

the 13th century by afiy-y-a-d-Dn al-Urmaw , in his Book of cycles.25 There, the smallestconceptual interval is the limma. The tone, is made up of two limmata and one comma, both

Pythagorean.26 The limma is equated27 to the semi-tone. Therefore, a typical tone may take the

form L + L + C, where L stands for thelimma, and C for thecomma. Therefore a pitch can beplaced in a scale within the limits of these intervals.28 In this case, the limma, and

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The comma play the role of elementary intervals (they are used to make up other intervals

in the scale). However, the comma is not a conceptual interval because it is never used as such

between neighbouring pitches of a scale but only as part of another and larger conceptual

interval. The comma and the limma, make up conceptual intervals used in the composition of

other intervals such as the neutral second, called mujannab which, according to Urmaw , canbemade up of two limmata (i.e.,L + L) or with one limma plus one comma (i.e.,L + C or C + L).

The difference between the two neutral seconds, i.e., the difference between two limmata

and one limma plus one comma (fig. 5: 177), or [(L+L) - (L+C) = (L-C)], is about 67 cents,almost three Pythagorean commata. Conceptually, however, the two possible forms of neutral


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seconds, with Urmaw , are equal. Both are called mujannab and considered as intermediateintervals placed between the limma and the tone.

Arabian theory has hardly changed since Urmaw .29 Modern scholars give two principalrepresentations of a scale with all possible locations of pitches.

The first is an approximation of the general scale with Holderian comma, HC,

henceforth, and the second uses the quarter-tone for quantification.

A HC equates to 1/53rd

of an octave, about 23 cents (22.6415)30

. Therefore one limmaequals to four HC, about 91 cents. This is close enough to the Pythagorean limma. The tone is 9

HC, or 204 cents, matching the Pythagorean tone. Typically, a tense diatonic genus31 is modelled

as a succession of two Pythagorean tones of 9 HC each, plus a limma with 4 HC tones. The

mujannab of Urmaw , whichamounts to a neutral second, has two possible values, 6 HC or 7HC, but they are considered as identical conceptual intervals.32

The first division of the octave, the 53-ET giving the Holderian comma as a common

divisor of all conceptual intervals, follows, in Arabian theory, complex rules that are given

elsewhere.33 The second division of the octave, in 24 theoretically equal quarter-tones, will

demonstrate a privileged example of interval relationship.

At this point, it may be useful to explain how two intervals, which are different in size,

can, according to Urmaw , be considered as identical conceptual intervals.34 The best exampleis with the maqm Bayt. It is based on the same scale as the maqm Rst. TheRst scale iscomposed of approximate three one-tone and of four three-quarter-tones intervals.

It could be noted as c d e- f g a b- c, with e- and b- being approximately one quarter-tonelower than their western equivalents. The scale of the Bayt is close to thegeneral structure ofmaqm Rst, but begins with d andhas a b flat (fig. 6: 178).

This gives d e- f g a bb c d.35The note e-, which has the same name in all theories of themaqm,36 is placed differently according to the context of the performance, or depending on thelocal repertoire (fig. 7: 179).37

In this maqm, the position of the degree sk hasa lower pitch in Lebanese folk musicthan it has in Classical Arabian music in the Near-East. Should we decide to use a quarter-tone

approximation for the intervals in Arabian music, as most modern theoreticians do, then the twoneutral intervals between d and e- and between e- and f are conceptualised as two three-quarter-

tones intervals fig. 6: 178). However, with the Dalna, in maqm Bayt, Near-Eastern folkmusic has a lower e-, which, regardless, is considered as a sk, but the lower interval between dand e-, the lower mujannab, is smaller than an exact three-quarter-tone (fig. 7: 179), and the

higher interval between e- and f is larger.38

Furthermore, the positioning of the sk dependson which maqm is played as well asregion and repertoire. A good example is in the difference between the position of the sk in themaqm Bayt and the position of thesame note in the maqm Rst which in this case is higherin

pitch, but lies approximately around the three-quarter-tone boundary. In the maqm Sk,39 orone of its frequent variants, the maqm Sk-Huzm,40 the position of sk is still higher andcould sometimes reach the upper value of Urmaws greater mujannab. This is the positionassigned to this note in modern Turkish theory.41

The boundaries for these different positions for sk are not established in practice, andthe study of its variations would require another paper.42

This pitch is perceived as a sk anywhere the player may perform. The difference isquantitative. However, the relative positioning of the note which is placed between eband e, will

always be perceived as a sk. There-fore, the conceptual understanding of the neutral second isnot simply quantitative, but also relative and qualitative. Importantly, the mujannab is perceived

as an intermediate interval between the one half-tone and the one-toneintervals. This appliesfor all other intervals such as the semi-tone which is an interval smaller than the mujannab, and

to the one-tone interval which is larger than the latter. The tonometric value of mujannab mayvary,43but it is the relative position of the interval in the scale and its qualitative and relative


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size, compared to other conceptual intervals, which gives it its full value in the repertoire. To

conclude on the nature of intervals in a scale, they are of three types:

1. An interval of measurement is an exact or approximate divider of other intervals. As a

general rule, any musical system based on the equal division of the octave, as in an equal

temperament, gives an interval of measurement, such as the semi-tone in the 12-ET, and with the

quarter-tone in the 24-ET or the HC in the 53-ET divisions of the octave.

2. Conceptual interval. This is one of the consecutive

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intervals of the second forming a musical system. For example, three seconds in a just

fourth, four seconds in a just fifth, or seven seconds in an octave. Conceptual intervals can be

measured either exactly or approximately with smaller intervals, usually of measurement, as in

approximations using the quarter-tone or the HC.

3. Elementary intervals are used in combination to build up to consecutive conceptual

intervals of seconds within a system. They can combine either with a similar elementary interval,

such as with the two limmata in Urmaws general scale, which combine into a mujannabinterval, or with another elementary interval, such as the limma +comma, for the second form of

mujannab, with Urmaw .These three types of intervals are not mutually exclusive. When the smallest conceptual

interval is also the smallest common denominator of all conceptual intervals, as with the semi-

tone in the 12-ET, then it becomes an interval of measurement, but it is also an elementary inter

val, although it remains conceptual when used as an inter val of second within a musical system.

The need to differentiate these three types of intervals arises within unequal temperaments, for

example with Urmaw.44This distinction will provide with a better under-standing of the combination processes

applied to music intervals.

Applying the concept of qualitative

differenciation of intervals on

Urmawis scale

Urmaws explanations about his scale show that the tone is composed of threeelementary intervals and that no interval within the fourth may contain either three successive

limmata or any two successive commata (fig. 8:180).

The comma is neither a quantifying interval as it does not divide exactly other intervals

such as the mujannab or the tone,45 nor is it a conceptual interval, as it is never used as a melodic

interval between two pitches in a modal scale.46 Furthermore, a comma is never used as the first

interval of a combination, with a notable exception for the mujannab which can hold the form

C+L.A conceptual interval generally starts with itself or with another conceptual interval. The

limma, for example, is used both as a conceptual interval, the smallest interval used in any of

Urmaws modal scales and as an elementary interval used in the composition of other, relativelylarger, conceptual intervals. With Urmaw , both the comma and the limma, are elementaryintervals. However and additionally, the limma is also a conceptual interval.

In modal construction, and with an appropriate choice of pitches within the scale, with

Urmaw , there areother conditions to be met. These include, for example, the inclusion of the

fourth and of the fifth. They must be complementary in the octave. With such limitations, wecan conceptualise the intervals of adjacent seconds in Urmaws modes in the following way (fig.8: 180):


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1. A conceptual interval of one semi-tone is composed of a single interval, part of the

scale. Since the smallest conceptual interval is the limma, we may conclude that the semi-tone is

equivalent to a limma.

2. The mujannab, or neutral second conceptual interval is composed of two elementary

intervals of the scale: the mujannab can be either composed of one limma +one comma, L+C, or

of two consecutive limmata, L+L. It is the only interval with Urmaw , listed among intervals

smaller than the fourth which may have two different sizes. As a corollary to this, two mujannabmay follow each other, but only if they have a different composition such as when one is L+C

and the other is L+L (or L+L then C+L).47

3. The tone is composed of three elementary intervals. However, a) three limmata must

not follow each other.48 b) The comma must always be preceded or followed by a limma. In this

case, the tone can only include two limmata and one comma, with two possible arrangements:

L+L+C, or L+C+L.

4. The greater, or augmented conceptual interval of the tone is composed of four

elementary intervals. It can only be made up of three limmata and one comma. They can only be

arranged in this manner: L+L+C+L or L+C+L+L. This interval is not mentioned in the Book of

Cycles. It is only assumed as part of Urmaws seconds.

5. The greatest conceptual interval of the second is made up of 5 elementary intervalsbecause the fourth can only be composed of a maximum of seven elementary intervals, within

the systematic general scale. However, two other intervals of second (conceptual interval) are

needed for its completion. Since the smallest second is the semitone, the limma, the greatest

conceptual interval is equal to the remainder coming from the subtraction of two limmata from

the fourth. The fourth is composed of two tones and one semi-tone, i.e., 2x(2L+C)+L, or 5L+2C.

Taking away two limmata, the resulting capacity of the greatest conceptual interval in a fourth is

3L+2C. Applying the rules of construction of the intervals, such as no more than two limmata in

a row, etc., the possible forms of the greatest second, or tone, in Urmaw-type scales areL+L+C+L+C, or L+C+L+L+C. This interval is not mentioned as such in the Book of Cycles but

is also assumed.

The fourth needs a combination of smaller intervals so that their sum can add up to itscapacity in terms of elementary intervals. In order to simplify the process, we shall use a simple

handling of numbers equating to the conceptual intervals of the second with Urmaw :49

1. The semi-tone equals number 1, as one elementary interval is needed to compose this

conceptual interval.

2. Mujannab is given the value of 2 since two elementary intervals are needed to build it

up to a conceptual interval.

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3. The tone interval is given the value of 3 since it needs three elementary intervals.

4. The augmented tone has the value of 4 since it requires four elementary intervals.

5. The greatest interval of the second within a fourth has the value of 5 because it needs

five elementary inter vals.

Although having a quantitative function in terms of numbers of elementary intervals

which make up a conceptual interval, numbers 1 to 5 express the intrinsic quality of the interval:

its identification as a different conceptual interval from those represented with another number.

As a common rule, the fourth is made up of three conceptual intervals. In order to comply with

Urmaw, they must add up to seven elementary intervals.Reduced to their hyper-systems, we have the following:

a. 115, with 1+1+5 = 7(not in Urmaws Book of cycles)

b. 124, with 1+2+4 = 7


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(not in Urmaws Book of cycles)c. 133, with 1+3+3 = 7

d. 223, with 2+2+3 = 7

Therefore, in this case, a fourth may contain, either a) two semi-tones, 1, and one

greatest interval of second, 5, or b) one semi-tone, one mujannab, or neutral tone, 2, and one

augmented, or greater tone, 4, or c) one semi-tone and two intervals of one tone, 3, or d) two

mujannab, or neutral tones and one one-tone interval. The algorithm for these hyper-systems isstraight forward (fig. 9: 180).

To find the first hyper-system, (fig. 9: 180, first step) take the smallest conceptual

interval, 1 twice in this case, and then deduce the value of the third interval by subtracting the

quantitative value of the first two, which adds up to 2 elementary intervals, from the value of a

fourth, or 7 elementary intervals, which gives 5.

The second hyper-system, the 124 hyper-system above, is obtained by decrementing the

value of the last digit interval in the preceding first hyper-system, (fig. 9:180, second step) and

by incrementing accordingly the value of the interval standing just before in the series: the last

digit in the first hypersystem is 5, which is decremented to 4, and the interval which precedes it,

which is the central 1 in the 115 hyper-system, is incremented, accordingly, to 2.

The simultaneous decreasing of one interval value by one unit, or its decremention, withthe increasing of one other interval value by the same unit of one, or accordingly incrementing it,

insures that the sum of the numbers in the series remains unchanged. Here it is equal to 7.

Applying the same process to the resulting hyper-system 124, (fig. 9: 180, second step

repeated) the third hyper-system is now 133. Applying the same process to this last hyper-system would result in 142. The capacity of this series is, however, the same as for 124. The

reason is that in the preceding 133, the last two intervals were equal but with the continuation of

the process in the same way, interval values for the central three are the same as the preceeding

values for the last three, i.e., 4 and 5, and reciprocally, which would result in the same

composition of intervals, in terms of quantity, within the fourth. At this point, we need to

improve the algorithm in order to find the remaining hyper-systems. This is done by decreasing

the rank of the intervals to be modified by applying the same process to the interval the rank ofwhich is immediately below the rank of the interval to which the decrementing process was last

applied, i.e., 133. The latter is the third interval in the series and now we must decrement the

second interval in the series, and increment, accordingly, the preceding one, the first interval in

the same series. Applying this process to 133 which we found in the preceding step, the second

interval, central 3 (fig. 9: 180,3rd step) is decremented to 2, and the first interval, 1, is also

incremented to 2, whilst the third interval, which is the last 3, remains unchanged. This gives the

new figure of 223. This is where the generation process ends since the two first intervals have

now similar values. Any further step would generate a redundant hyper-system.50

Now that we have determined the hyper-systems agreeing with Urmaw, we need extractall possible genera and shades to give the full range of intervals in the fourth. The next section

will review combination processes of intervals, for any hyper-system.

Various forms of interval combinationThere are different methods for combining intervals. One is the rotation and

permutation process. It is the most common. Rotation was used, notably, by Aristoxenus in his

Elements of Harmonics,51and permutations were often used throughout history, and most

probably by Frb in hisgenera, adding to Aristoxenus range of tetrachords.52 Both processesare deficient since they do not give, in their simplest expression, a full account of all the possible

combinations. The tree process given below has the whole range of results. However this is more

related to statistical and probabilistic analyses.

There are other procedures, such as de-ranking, which can be considered as a generalcase of the Byzantine-wheel method. Modal systematics uses them all for the purpose of

arranging and classification, with special recourse of the de-ranking process.


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Rotation of intervalsRotation (fig. 10: 181) is a straight forward process by which intervals may be combined,

placing the first after the last one, or inversely, the last before the first, leaving the other intervals

in their position.

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The first method is a clockwise process which continues as long as the first interval does

not come back to its initial position, obviously. Figure 10, page 181, shows that this process

generates intervals in three different ways (the first does not rotate since it places the interval

system in its original and basic position). However, the rotation process is defective, as it always

gives three possible combinations of three intervals, whenever the combination possibilities for

these three intervals allows for six different combinations.53 For the purpose of his explanation,

Aristoxenus used intervals of the enharmonic genus which are made up of two quarter-tones and

one di-tone, that is two equal intervals out of three. Figure 11, page 181, shows intervals with

subscript numbers so that they retain their initial rank in the basic configuration, that is a1 as the

first interval of the basic configuration, a2 as the second and b3, as the third. Even then, the

rotation process gives three distinct combinations. If the three intervals are equal to Frbsequal-tone distribution where each is 5/6 of a tone, a combination process, whatever it may be,

will always give the same result as combining the three intervals a a a.

Other processes are more effective but Aristoxenus use of this limited process mighthave been a consequence that he considered interval combination as a de-ranking process.

Permutation of intervalsPermutation exchanges one interval for another whilst others remain fixed. The same

process is applied to another pair until all intervals have changed places.With direct permutation (fig. 12: 181), interval a1 of the basic configuration a1 a2 b3 is

first changed with inter val a2. This results in combination a2 a1 b3. Then, coming back to the

original configuration, with b3, which is the combination of b3 a2 a1. As a1 has already changed

places with the two other intervals, we procede with the second interval of the basic

configuration, a2, with the others. This interval has already changed places, in the previous

process, with a1: it should further change places with b3 only with the combination a1 b3 a2.

The last interval has already changed places with both other intervals a1 and a2, and this is

where the process ends.

If a1 is different from a2, and also from b3, then the second combination: a2 a1 b3 is

different from the first combination, because it is a stand-alone interval system. The total number

of distinct interval systems which result from the direct permutation process is 4, that is onemore than with the rotational process. If both intervals are the same, however, if a1=a2, the two

first combinations are equal. The process only gives three different combinations, similarly to

those in the rotation process. In order to obtain the full range of possible combinations for these

three intervals, we could apply the process of direct permutations, not only to the original

configuration of a1 a2 b3, but also to each of the combinations which result from the direct

combinations of a2 a1 b3, b3 a2 a1 and a1 b3 a2. If we apply this process to the second in the

direct permutation process, combination b3 a2 a1, we obtain the following combinations:

1. New base: b3 a2 a1. This is the second combination in the direct permutation process.

(fig. 12: 181)

2. Combination N2: a2 b3 a1, consisting in exchanging the first interval with the second.

This is a new combination, different from all the previous ones.


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3. Combination N3: a1 a2 b3, consisting in exchanging the first interval in the new basic

configuration with the third one. This gives the same combination as the first one in the direct

permutation process.

4. Combination N4: b3, a1, a2, by exchanging the second interval in the new basic

configuration with the third interval of the same. This is also a new combination.

Therefore we have two new interval combinations which added to the four distinct

combinations of the direct permutation, amount to six different combinations of the threeintervals a1, a2 and b3. These amount to the possible combinations with three distinct intervals.

There is no need to apply the permutation process for the other combinations stemming from the

first direct process. It is also possible to obtain a similar result with processes other than the

successive permutations method, for example by applying rotation followed by a direct

permutation process (fig. 13: 181). In this combination process, a direct permutation process is

applied to each of the combinations coming from an initial rotation process (fig. 11: 181). This

gives six independent and distinct combinations out of twelve. The six remaining combinations

are redundant.

Let us be reminded of two characteristics of the reviewed combination processes:

1. The successive, or consecutive permutations and the alternate rotation/permutation

processes generate a certain number of redundant combinations which have to be excluded fromthe outcome.

2. Out of six distinct resulting combinations obtained, three will be redundant if a1 equals

a2. In this case, the outcome remains the same as for a simple rotation process (compare with fig.

11: 181).

Tree processing54Here, in the tree processing the combinations are based on an initial choice of intervals,

rank by rank (fig. 14: 183). With the first rank, we may chose between the three intervals a1, a2,

or b3 (the subscript plays here more the role of identifier for each interval, than the role of an

initial rank number).Having completed this first step, we still have two intervals of which one must be

assigned to the second position in the series. The third step leaves us with one possibility since

two out of three intervals have already been used.

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The process is straightforward as it gives directly the six distinct combinations seen

above. There are no redundancies although intervals a1 and a2 could be taken as equal. In this

case, again, we only have three distinct combinations.

The tree processing method is rarely used for combination of intervals and this is one ofthe reasons why we have to explore further the de-ranking process which is of crucial importance

in modal systematics55as it is a practical way for arranging and classifying large numbers of

interval combinations, such as in the heptatonic scales.

The de-ranking process, or picking intervals

N in a row out of repeated series of Mconjunct intervals - Hyper-systems, systems,

and sub-systems.De-ranking is closely related to rotation. It is very useful and in the study of musical

systems applies mostly to the double octave. In a reduced form, the de-ranking process takes it

that a series of conjunct intervals is repeated a certain number of times, for example for in the

series a1 a2 b3 a1 a2 b3 a1 a2 b356 By de-ranking the first interval, we start the series of


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intervals by the first interval a2 instead of the first interval a1. We may consider this process as a

rotation of intervals where the first a1 goes to the end of the extended series. If we choose N

intervals out of a repeated pattern of N intervals, this process is a repeated rotation where N = M

= 3. (fig. 15: 183)

In a more general application of this process, N intervals in a row are taken out of a series

of M, repeated at least once, with both N and M being integer numbers. In the case of five

intervals a b c d e repeated once in a row, for example (fig. 16: 184), we can pick up any seriesof three conjunct intervals to form a combination. The first ranking combination is a b c, the

second b c d, the third c d e, etc.

If we apply this process to a double heptatonic tense diatonic scale, and in turn select

seven conjunct intervals among the fourteen of the series (fig. 17: 184), beginning with the first

interval, the second, the third, etc., and until the seventh, we obtain the seven different species of

the scale.57In figure 17, page 184, the basic scale is 1 2 2 1 2 2 2, in which intervals are

expressed as multiples of the semi-tone. This corresponds to the diatonic, and here also, the

equal temperament western scale beginning with B or its equivalents (b, b, etc.), or B 1 (semi-tone) c 2 d 2 e 1 f 2 g 2 a 2 (b). Of all possible species of the double diatonic octave, this scale

corresponds to the lowest value when expressing the concatenated intervals as an integer

number.With modal systematics, the first in a series of deranked combinations is considered as

the basic system.58

The others, in this example, are sub-systems of system 1 2 2 1 2 2 2 (fig. 18: 185). The

hypersystem is the interval capacity indicator that we find in arranging all intervals in a

combination from the smallest to the largest. For example, in the hyper-system 1 1 2 2 2 2 2,

from which we get that the capacity of all corresponding systems and sub-systems is equivalent

to two one-semi-tone intervals and five one-tone intervals, there are other systems which are

distinct from 1 2 2 1 2 2 2. They have the same capacity, within the same hyper-system. In order

to find all systems and sub-systems originating from a hypersystem, one needs apply, for

example, a combined process of rotations/permutations to its intervals.59This has been explained

above60. If we eliminate the redundant systems or sub-systems, we find two other systems forhyper-system 1 1 2 2 2 2 2. The first of these two distinct systems is the hyper-system itself, as it

expresses an arrangement of intervals 1 1 2 2 2 2 , which is different from 1 2 2 1 2 2 2, where

the two semi-tones in the first combination are placed in a row. This system has in turn seven

sub-systems. In this case, they are species. The remaining system which has the same interval

capacity as the precedent ones but whose intervals are arranged following a different pattern

where two semi-tones are separated alternately by one, then four, one-tone intervals, is 1 2 1 2 2

2 2, and has, accordingly, seven distinct sub-systems. Figure 19, page 185, shows how hyper-

system 1 1 2 2 2 2 2 has intervals that can be combined in three distinct systems which in turn,

give seven different combinations or sub-systems obtained from de-ranking.

This hyper-system is peculiar in that it is the only one composed exclusively of one

semi-tone or one tone intervals. If to our alphabet of intervals, we add the one-and-a-half-tones interval class in our model, we find twoother hyper-systems, 1 1 1 1 2 3 3 and 1 1 1 2 2 23. These generate 15 and 20 distinct systems, respectively, or 105 and 140 distinct sub-systems.

They are too numerous to be listed here, but an example of sub-system from the first hyper-

system is the scale of the well known Hijz-KrArabian mode, with two Hijz tetrachords (1 31),61 separated by a one-tone interval: 1 3 1 [2] 1 3 1.62 Another example, related to the second

hyper-system, is the scale of the contemporary Arabian maqm Hijz, which commonly followsthe scale 1 3 1 2 1 2 2 when reduced to a semi-tone scale without neutral intervals.

Now if we wanted to express the intervals of these hyper-systems in the equal-quarter-

tones distribution of modern Arabian theory, then this would give:

2 2 4 4 4 4 463

2 2 2 4 4 4 6 2 2 2 2 4 6 6


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If arranged in agreement with modal systematics classification, with the lesser values of

hyper-systems holding the lower rank, their places would be reversed as:

157

1. 2 2 2 2 4 6 6

2. 2 2 2 4 4 4 6

3. 2 2 4 4 4 4 4

Let us now take in consideration the two neutral intervals used in modern Arabian theory.

These are the three-quarter-tone 3 and the five-quarter-tone 5 intervals, which areconceptually differentiated from the one-semi-tone, one-tone, and one-tone-and-a-half.

Combining the five intervals 2, 3, 4, 5, and 6 in seven possible positions, with the condition that

the sum of the intervals must be equal to 24 quarter-tones, we end up having 19 hyper-systems

(tab. 5: 175) with a possible number of 4795 sub-systems or scales. Among them, there are very

few in usage. Scales used in semitone hyper-systems such as hyper-systems no. 1, 6 and 12 inthe table, are limited to the diatonic and to the ijz-Kr orijz type scales. For theremaininghyper-systems, scales used in the performance, practice and theory of Arabian, Persian and

Turkish64 music are remarkably few, no more than 150 to 200 when compared to the possible

number of 4975, or out of more than eight thousand possible sub-systems with the extended

alphabet (i.e.,with intervals greater than the one-and-a-half-tones), as we shall see in Part II.65

Some preliminary remarks on the systems and sub-systems of the quarter-tone generative

model can already here be expressed:

a) Homogeneity of interval composition within a hyper-system results in a lesser number

of systems because of the redundancy factor. The less the interval contains different classes of

intervals, for example hyper-system no.12 contains only two classes of intervals, the 2 and the 4,

the less it generates systems and, consequently, sub-systems.66b) Two relatively homogeneous hyper-systems, no. 16 = 2334444 and 19 = 3333444,

generate scales which are mostly used in Arabian music. Hyper-system no. 17, al-though very

homogeneous, 3333336, is not in use because its intervals can add up neither to a fourth

(sum=10) nor to a just fifth (sum=14).

c) Hyper-systems nos 1, 6 and 12, share with hypersystem no. 19 an important feature:

more than half of their sub-systems have a fourth or a fifth beginning with the first interval.

d) System 2 4 4 2 4 4 4 in hyper-system no. 12 (this is the diatonic system that we have

noted before) maximizes the number of fourths or fifths since six out of seven of its sub-systems

contain a direct fourth and a direct fifth in relation to the tonic. Seven out of seven have either of

them. This is the only system, among those generated with this model, with such qualities.

e) Hyper-systems which have the augmented seconds of Western music in a ijztetrachordal combination (i.e.,containing at least one interval of one-and-a-half-tones or 6 and two intervals of one-semi-toneor 2in the form 2 6 2) generate large numbers of systemsand sub-systems; these are hyper-systems no. 1, 6 and 9. This is an indication that these scales

are a reservoir for modulation from and to diatonic scales.

f) Along with hyper-systems no. 12, 16 and 19, these generate about one hundred sub-

systems that are the most frequently used or mentioned in specialized literature. These scales,

although stemming from hyper-systems with a reduced generative capacity, with about 22% of

the total of sub-systems, form from two thirds to three quar-ters of the reservoir of scales used, or

attested in Arabian music.67 Their ratio of sub-systems with a double fourth and fifth from the

tonic is close to 39% with most of the other sub-systems in usage (see rows with variants orclose to in the column of remarks of table 5, i.e., hyper-systems nos 4, 10, 11 and 15 the


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number of sub-systems marked FF for these represents a ratio of more than 46% of the total)

contained in hyper-systems related to them.

g) In the Remarks column with tab. 5, variants are mainly scales containing analternative ijz tetrachordmade up of intervals of 2, 3 an 5 quarter-tones. This is a possibleindication that this tetrachord evolved from earlier forms such as 2 5 3 or 3 5 2, to our

standardized form of 2 6 2, because of the pressure induced by the existence of the semi-tone

equal temperament.68

These remarks, made on the basis of the quarter-tone generative model of modal

systematics, suggest already some criteria which may be applied in statistical studies of systems

and sub-systems as we shall apply in Part II of the present paper.

These criteria will help answer the question why out of 12 possible intervals in a semi-

tone distribution, or out of almost 24 intervals in a quarter-tone distribution, only seven are

combined, in most music, to form an octave? And why are there three intervals in a fourth and

four in a fifth, generally.

Before answering these questions, we must return to Urmaws genera, in order to have abetter understanding of how, by applying the qualitative interval differentiation concept, uneven

divisions of the octave can amount to even ones.

Applying modal systematics to

Urmaws generaIn Urmaws model, we have distinguished intervals of the second by means of the

capacity of integers, from 1 to 5 (fig. 8: 180). If we combine these intervals in the frame of a

fourth, the sum of which must be equal to seven elementary intervals, we obtained the following

hyper-systems:

1 1 5 1 3 3 1 2 4 2 2 3

158

Hyper-systems within the fourth as a containing interval, with two identical intervals

generate one single system equivalent to the generative hypersystem. They amount to three: 1 1

5, 1 3 3 and 2 2 3. Among them, the last two agree with Urmaw in theBook of cycles, withintervals not greater than the tone. By de-ranking, possible combinations of the intervals

contained in the three aforementioned hyper-systems are, for the first, combinations 1 1 5, 1 5 1

and 5 1 1. For the second, combinations 1 3 3, 3 3 1 and 3 1 3. For the third, combinations 2 2 3,

2 3 2 and 3 2 2. The remaining hyper-system, 1 2 4, generates two systems resulting in six

distinct combinations which stem from 1 2 4: 1 2 4, 2 4 1 and 4 1 2, and stemming from system 1

4 2: 1 4 2, 4 2 1 and 2 1 4 (fig. 20: 186, left).

All genera in hyper-systems 2 2 3 and 1 3 3 are known both to Urmaws Book of cyclesand to modern maqm theory of the quarter-tone division of the octave.(fig. 20: 186,

right).

The possible and missing genera in the treatises have in common peculiar features: each

of them contains two small intervals in a row, either two consecutive conceptual semi-tones or

limmata, or a limma and a mujannab in a row, similar to the 1 and 2 intervals in Urmawsqualitative model (fig. 20: 186, left), and the 2 and 3 quarter-tones intervals in the quarter-tone

model (fig. 20: 186, right). This is another criterion which will be applied in the statistical study

which follows.

At this point, we may also note that the connection between the quarter-tone model and

the model in Urmaws qualitative interval equivalents is straight forward: in order to shift fromUrmaws model to the quarter-tone model, add one unit to each interval in the first. (tab. 6: 176)All the scales of the quarter-tone model connect directly with Urmaws qualitativerepresentation, through a unitary vector subtracted from the interval values in the former. For


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example, the maqm ijz scale, 2 6 2 [4] 2 4 4 (sum =24) in modern maqmtheory (the squarebrackets identify the disjunctive tone between two tetrachords), becomes 1 5 1 [3] 1 3 3

(sum=17) in Urmaws model, and the maqmRst 4 3 3 [4] 4 3 3, in quarter-tones becomes 3 22 [3] 3 2 2, or two similar tetrachords composed of, successively, one one-tone and twomujannab intervals, with a disjunctive one-tone [3] interval.

In the model applied to Urmaws intervals which consist in a division of the octave in 17

equal parts, the total sum of the intervals must amount to 17 elementary intervals in one octave.The transition to the quarter-tone interval is straightforward, as by subtracting one unit in each

conceptual interval of a heptatonic scale in the quarter-tone model, we end up subtracting seven

units from the total of 24 quarter-tones, which gives the sum of 17.

All the scales of the quarter-tone model, arising from the hyper-systems in table 5, page

175 have equivalent counterparts in Urmaws model,69which proves that thetwo models are, inessence, conceptually equivalent.70 As a further consequence, all the results from the statistical

analysis, resulting from generations with the limited alphabet, from 2 to 6 quarter-tones, may be

applied to Pythagorean equivalents in Urmaws model.71 Another conclusion may be drawn atthis stage. Urmaws concept of the scale,regardless of Pythagorean procedures used to explain,or legitimize his ideas about music, is profoundly Aristoxenian and based on a combination

model. This applies as a rule to the composition of conceptual intervals using elementaryintervals. The intervals within a fourth are derived from a combinatory process where the fourth

and the fifth add up to an octave.72

Conclusion for part IA quantitative model based on the equal division of the octave can be a qualitative

model, taking in account the size of the intervals of which the scale is composed. They express

the number of elementary intervals which build up each of the conceptual intervals. In the case

of the quarter-tone model, the smallest elementary interval is the approximate quarter-tone (the

measuring interval), the smallest conceptual interval is composed of two elementary intervals, or

two approximate quarter-tones, etc. Combining the resulting conceptual intervals, we combinequalities of intervals that are differentiated by their capacity to contain elementary small

intervals73(fig. 21: 186). This means that the scales which result from that type of generative

model have intervals of seconds which, if measured exactly, would differ from one another even

when having the same interval capacity (a one-tone interval in one scale may be slightly different

from a one-tone interval in another scale).74 However, these intervals, when taken in relation to

other intervals in the scale carry a unique quality which differentiates them from the latter, which

is typical of modal systematics.

159

Part II. Combining intervals in a system:

statistical analysisWith modal systematics the basic process consists in combining intervals

expressed as integers and then analysing the results in relation to both music practice and

theory. The elements of the scale consist in a sequence of consecutive conceptual

intervals.

Conceptual intervals are stand-alone units in the scale. They are distinct in theory and in

practice. They are placed between the notes of the scale. Their function is qualitative.75 For an

immediate identification of any interval in a scale series, modal systematics determines the

optimal (or the smallest, with the largest elementary interval) division of the scale, in such a waythat the quantifying interval is the smallest conceptual interval and the elementary interval. In the


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semi-tone scale, the semi-tone is such that it fulfills the functions of quantifying, elementary and

conceptual intervals.

With Arabian music76, the semi-tone model is ineffective because conceptual intervals,

such as the neutral tone or neutral augmented second the mujannab and the greater tone inUrmaws model in fig. 8, or the three-quarter-tones and the five-quarter-tones intervals in thequarter-tone model (fig. 21: 186), cannot be distinguished and identified as conceptual intervals.

Therefore, another division of the octave is necessary to provide qualification for all types ofintervals. In this case it is the 17-ET, or the division of the octave in 17 equal intervals77 which is

needed, since this division allows for the distinction of all conceptual intervals. These small

intervals have values (fig.8: 180) of 1 to 5.

Integers segregate the semi-tone 1, the mujannab or neutral second 2, the tone 3, the

neutral augmented tone, or greater tone above 4 and the fully augmented tone, greatest tone

above 5.78 However, the 17-ET model has a flaw which makes it difficult to see it as a

representative division of the octave. If taken strictly as a measuring interval, the 17 th of an

octave is 71 cents (fig. 22: 187). Adding these intervals, we have 494 cents for a fourth and 706

cents for a fifth. These figures are close enough to the corrected values of the fourth and the fifth

in the Pythagorean system, i.e., 498 cents and 702 cents. The problem lies with the

representation of the semi-tone. If the 17thof an octave is conceptualised as a semi-tone interval,the discrepancy with an equal temperament semi-tone, in approximation is 29 cents, or 100-

71=29, which is unacceptable to most musicologists. As a result, and although the measuring

17thof an octave interval which divides the octave in 17 equal parts is also an elementary 79 and a

conceptual interval,80 we shall take the quarter-tone model for Arabian-Persian-Turkish music

bearing in mind the equivalence between the two models.81

The principle of economy: optimal balance between method and expression In his first

paragraph of his Tonality of homophonic music, Helmholtz said of the musicians liberty that:Music was forced first to select artistically, and then to shape for itself, the material on

which it works. [...] Music alone finds an infinitely rich but totally shapeless plastic material inthe tones of the human voice and artificial musical instruments, which must be shaped on purely

artistic principles, unfettered by any reference to utility, as in architecture, or to the imitation ofnature as in the fine arts, or to the existing symbolical meaning of sounds as in poetry. There is agreater and more absolute freedom in the use of the material for music than for any other of the

arts. But certainly it is more difficult to make a proper use of absolute freedom, than to advancewhere external irremovable landmarks limit the width of the path which the artist has to traverse.Hence also the cultivation of the tonal material of music has [] proceeded much more slowlythan the development of the other arts. It is now our business to investigate this cultivation.

82For thousands of years, freedom in music has been restricted by the necessity to produce

recognisable pitch patterns making up melodies.83 To this end, most cultures use heptatonic

scales. They are a paradigm for composition. In order that a melody can be recognised, the

degrees of the scale must be identifiable by pitches in relation to the other degrees of the scale.

When these are expressed as intervals, they become conceptual intervals where each has its ownquality so that they can be identified. Conceptual intervals must neither be too small as they

would be too difficult to perceive, nor too big, as in both cases melodies may not be easily

perceived.

Variations of intonation or subtle differences of intervals, especially with music which

does not answer to any known temperament, are the consequence of impromptu performance,

great mastery, regional variations, organology, particular tuning and so forth, all combined with

the ability of the performer.84 In a traditional process of knowledge transmission, however, these

subtle variations, particularly in the domain of performance mastery and instant creativity, take

place at a later stage of music understanding and perception. In order to transmit and receive, a

basis must be found allowing for a firm structure of the musical discourse, whilst allowing the

performer the possibility to further develop his freedom of interpretation. This basis, which is theessence of the melodic repertoire, is commonly named the scale.


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When confronted with an audience, a traditional musician of average talent would try to

perform with utmost expression and invention, keeping in mind the need for a melodic pattern

that his listeners will recognise. This process should request the least possible energy whilst

taking the least possible steps within the continuum of pitch, in the search for balance between

technique (or complexity

160

of the means used) and expression (or the effect of the musical discourse on the

audience). In order to achieve this goal, this musician would ideally need to have previously

tested all possibilities, within an octave or other important containing intervals, and determine

which would result in the maximum number of expressive possibilities.85 The process for interval

combination and the search for the optimal number of intervals within a fourth, a fifth oranoctave86 , will be defined as stemming from the principle of economy.

A semi-tone and quarter-tone model for

genera and scalesThe two models in this study are the western and the Arabian. Whilst western music uses

the semi-tone, the quarter-tone is the basis of conceptual divisions with the maqmwhere subtlerefinements reveal modal complexity.87 In both cases, the smallest conceptual interval is an

approximate semi-tone. A recurrent objection to the use of the semi-tone interval as a smallest

conceptual interval is that the Arabian quarter-tone is half its size. Some theoretical modern

descriptions of maqm Awj ra, for example, show indeed one quarter-tone intervals in thescale. This maqm is reminiscent of Turkish music as its scale is similar to the maqmijz-Kr,but with awj (= b-) 88 as its starting note. This causes some cultural and technical problemswith the organology of the `d,because of the usage of the theoretical equal quartertone in the

1920s and 1930s.89

These problems are easily resolved with the difference between conceptualand measuring intervals we have explained.90

Another objection to Arabian performance is mainly with maqm Sk . It begins, as itsname suggests, with sk, = e-. In the quarter-tone model, this scale equates to 3 4 4 3 3 4 3,

beginning with e- and with a b- between the conjunct two-three-quarter-tones intervals. In the

common Arabian tuning of the d, 91 the open strings of lower pitch are often used as dronesrepeating the fundamental note of the maqm .92 In order to reinforce the fundamental, somecontemporary lutenists tune the lowest auxiliary string to E-. Many, however, prefer to keep the

original tuning93 and use a technique of fast alternation between e- and a note about a third of a

tone lower,94 and quickly coming back and insisting on e- so that the fundamental e- is

reinforced. The small interval used between e- and the slightly lower pitch is only a variation and

is used intonationally. Its main function is to underline the importance of e-, the next, the lowerdegree in the scale being d, which is three quarter-tones away from e-. This is why the performer

must use a smaller interval of intonation leading to the fundamental.

Now that these two main problems have been addressed and that the limitation of small

intervals is taken in both models as equal to the conceptual semi-tone, two possibilities have

been considered regarding the largest interval in the scale. It must be firstly limited only by the

size of the octave, and by the minimum of two intervals amounting to a scale element, or

secondly by the largest conceptual interval in both models, i.e., the one-and-a-half-tones

interval.95 As a result, each generative process uses alternatively two alphabets. In the semi-tone

generation, the first alphabet is without limitations except for the semi-tone division. The largest

interval in the alphabet is the largest possible allowed in a particular generation. The second

alphabet is reduced to the three conceptual intervals of one semi-tone 1, one tone 2, and one and

a half- tones, 3, or augmented second.


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This also applies to the generation process for the quarter-tone, except that in this case,

the interval increments are quarter-tones, with a limited alphabet of 2, 3, 4, 5 and 6 of them (fig.

21: 186). The generative process is simple. A computer programme detects all the combina-

tions of a certain number of intervals given in an initial alphabet of conceptual intervals, and

arranges the results as hyper-systems, systems and sub-systems. This process starts with the

minimum possible number of intervals in the scale elements96 and ends with the maximum

possible number of elements in the containing interval. The minimum number of intervals incombination is two, and the maximum depends on the containing capacity of the intervals in the

model. With both models this corresponds to the number of half-tones in a row which can be

arranged in a containing interval, i.e., five for a fourth, seven for a fifth and twelve for an octave.

Preliminary definitions and remarksSpecialised terms for scale systems will be used throughout this study, their definition

follows:

1. A scale system is a sequence of numbers for different classes of conjunct (conceptual)

intervals within the frame of a containing element. This is defined as an interval composed of

conceptual intervals with the sum of the containing element equating to the number of

elementary intervals building up to it set to a certain value. Containing intervals are equal to the

fourth, with an ascending frequency ratio of 4/3, and the fifth, with a frequency ratio of 3/2, and

the octave.

2. A hyper-system is a capacity indicator of conceptual intervals. It is a scale element in

which these intervals and the numbers composing the sequence, are re-arranged to form the least

integer when numbers are concatenated. Hyper-systems are arranged, in the frame of a

generative process, from the smallest (when expressed in integer concatenated form) to the

largest.

3. A system is a particular arrangement of intervals in a hyper-system. Systems are also

scale elements. They are arranged from the lowest corresponding integer to the highest within

the hyper-system. A hyper-system is identical to

161

the first ranking system it generates.

4. A sub-system is a particular arrangement of intervals inside a scale element which

corresponds to a de-ranked system. The original system is the first sub-system, and each de-

ranking produces the next ranking sub-system. The number of conceptual intervals, NI,

henceforth, limits the number of sub-systems in a system, as some of the combinations resulting

from the de-ranking process may be identical and therefore redundant. The number of non

redundant sub-systems may be lesser than the corresponding NI. The first ranking sub-system in

a system is identical to the head system.5. NI is the number of conceptual intervals of conjunct seconds which constitute a scale

element. In the statistical study below, NI is variable and extends from two conceptual intervals

in a scale element, to the maximum possible number of smallest conceptual intervals in a row

within the containing interval. In both models, the maximum number of conceptual intervals in a

scale element is equal to the number of conjunct semi-tones - the smallest conceptual interval -

required to build it up. The maximum number of conceptual intervals in a containing interval

(NImax) of a fourth is equal to the number of semi-tones needed,i.e., five consecutive semi-tones

(NImax =5). A typical ex-ample of the relationship between hyper-systems, systems and sub-

systems is shown in figure 19, page 185, and table 5 where the 19 hyper-systems of the quarter-

tone model generation with the limited alphabet 2, 3, 4, 5, and 6, and with seven intervals (NI =

7) to the octave, are arranged in ascending integer values. A typical hyper-system generatesdiatonic scales,i.e., hyper-system no. 12 in the generation with the reduced alphabet (tab. 5: 175).

This hyper-system generates three systems (fig. 19: 185) for the corresponding semi-tone model,


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when each in turn generates 7 distinct sub-systems by de-ranking intervals in each system. 97

Table 5 is specific to the general combination process used in modal systematics. Since the

containing interval is equivalent to the octave, the sum of the integers (in unconcatenated form)

in each scale is 12 half-tones in the semi-tone, and 24 in the quarter-tone model.

With the fourth, the respective sums in the two models are 5 semi-tones or 10 quarter-

tones, and in a just fifth 7 and 14 respectively. The equality of the intervals of the semi-tone and

the quarter-tone models is straightfor ward. For the transition from a semi-tone interval system toits equivalent in the quarter-tone model, simply multiply the intervals of the integers by two. To

reverse the process, divide all the integers in the quarter-tone model by two. However, intervals

represented by odd integers in the quarter-tone model have no equivalents in the semi-tone

model. This is the reason why the ranks of the hyper-systems in the semi-tone model are

corrected to their rank in the quarter-tone model, as explained in the next section.

The main question is why the generally assessed number of conceptual intervals in a

modal scale is seven in an octave, or what is the optimal number of conceptual intervals in

containing intervals with ratios 4/3, the fourth, 3/2, the fifth, and 2, the octave.

Combining intervals within a fourth:

filters and criteriaIn a combination process of conceptual intervals using the semi-tone as the smallestconceptual interval, the sum of the containing interval of the fourth 98 must be 5 in the semi-tone,

and 10 in the quarter-tone models. Our first goal is to find all combinations of intervals of the

alphabet that sum up to these values.

In the semi-tone generation (fig. 23: 188, top), the alphabet is unlimited, except by the

semi-tone structure of the intervals. The smallest interval is the semitone, and the largest, for

NI=2 (two intervals in combination) can therefore only be a 4 semi-tones interval, 4 in the

concatenated sequence of intervals, [14] in the first hyper-system of the semi-tone scalegeneration with NI=2.

The sum of the two intervals in the first hypersystem is equal to 1+4=5. The other hyper-

system for NI=2 is 23, with two intervals 2 and 3 (the semi-tone value is represented by the twodigits.)99 The rank of the hyper-systems (first column to the left) is given both in the semi-tone

(plain numbers) and the quartertone models (between brackets) if the two differ. If the hyper-

system does not exist in the semi-tone model, only the rank of the corresponding quarter-tone

hypersystem is given (one number between brackets). For NI=2, the two hyper-systems 14 and

23 both generate one single system, with two sub-systems for each system. For NI=3, we still

have two (but different) hyper-systems (or capacity indicators) which generate each one single

system, but with three sub-systems each (due to the three conjunct intervals in the system).

This generation corresponds to the commonly accepted number of three intervals in a

fourth, and contains the tetrachord equivalent of the tense diatonic genus, hyper-system 122, and

of the tone, or tense chromatic: 113. For each of NI=4 and NI=5, we obtain one single hyper-

system, with four sub-systems for NI=4, and five identical, (with four which are are redundant)sub-systems for NI=5. The total numbers of hyper-systems, systems and sub-systems in each

case figure in the row below the last sub-system.

A first, and evident remark can be made. A small number of intervals, NI, implies that

larger intervals have more chances to find a place in the system, whenever a larger NI results in

an increased use of smaller intervals, notably here the semi-tone. Additional rows below the

grand total give the numbers of remaining sub-systems for each NI whenever some eliminating

conditions are met (the number of excluded sub-systems is shown in brackets, with a minus

sign):

162

1. The total number of non-redundant sub-systems is equal to the initial total number of

sub-systems minus the number of redundant sub-systems in each case. Redundancy occurs once


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in the semi-tone model, for NI=5. Here the hyper-system, system and sub-system(s) are identical,

as one single interval class, the semi-tone, is used in the scale element. These redundant sub-

systems, generated through the de-ranking process, are struck out in figure 23, page 188, and

must be excluded from the generative process.

2. The double semi-tone criterion (an asterix is added at the end of each sub-systemwhich responds to this criterion) excludes (separately from other filters) sub-systems containing

two semi-tones in a row (conjunct semitones).100

This filter, which has been inspired fromArabian music, is most effective when applied to sub-systems with a large number of intervals of

greater values. If two consecutive semi-tones are present in a heptatonic scale, they are

commonly found at the sides of the junction between two tetrachords, or at the junction between

a scale and its equivalent to the octave, lower or higher.101 For larger containing intervals such as

the fifth and the octave, this criterion is applied for three conjunct semi-tones.102

3. The conjunct large intervals filter (sub-systems marked with ) excludes scaleelements containing at least two conjunct intervals larger than, or equal to, the onetone interval,

and among which one, at least, is larger than a tone. This is a general rule which is present in the

heptatonic Arabian traditional scales. Examples of sub-systems with such characteristics are 46

and 55103 for NI=2 in the quarter-tone model (fig. 23: 188, bottom). The criterion is most

effective with smaller values of NI.1044. All these filters operate independently. If we combine them in one complex criterion,

filtered subsystems will add up or merge (neither nor, or a Boolean inversed OR operator inthe theory of ensembles) with a resulting number of filtered sub-systems in the row entitled

Intersecting criteria 1.The aim is to compare, excluding all filtered subsystems, the results ofthe generative process for different values of NI and to determine the optimal number105 of

conceptual intervals in the containing interval. The results of the semi-tone generation, with or

without filters applied to them, are shown in the two graphics of figure 24, page 189. The

generation with NI=3 (or three conceptual intervals in a containing fourth interval) gives the

largest number of independent, non-redundant, sub-systems, i.e., 6. The filters or criteria,

accentuate this optimal value. If we exclude scale elements comprising large intervals (greater

than the one-tone-and-a-half)106 in addition to those excluded through the intersecting criteria 1composed filter, the remaining two sub-systems in the case NI=2 would be equally eliminated,

leaving thus the case NI=3 as the unique possibility concerning the ability to generate a just

fourth (see intersecting criteria 2, fig. 23:188 ).107 The same applies to the quartertonedistribution, (fig. 25: 190) with however some quantitative and qualitative differences in the

contents of the two generations.

A first difference is that the quarter-tone model (fig. 23: 188, lower half and fig. 25: 190)

generates, as expected, intermediate and additional hyper-systems containing neutral interval

equivalents (or odd multiples of the quartertone) which are for example hyper-systems nos 2 and

4 in the case of NI=2. Whenever the smallest and largest intervals are the same, in both semi-

tone and quarter-tone generations, for the same NI (due to the limitation of the smallest

conceptual interval to the semi-tone in both cases), then the intermediate hyper-systems generate

ad ditional sub-systems in the quarter-tone model. The optimal number of intervals (the most

economic choice) is still three. All the filters accentuate the optimal value by giving the two

neighbouring sections of the line a smaller angle (in fig. 25: 190, top, intersecting criteria 1give the most acute angle around value 13 for NI=3). Figure 23, page 188 shows, however, that

the new possibilities in the quarter-tone model are not fully integrated, for NI=3, in Arabian

music, although this case comprises no redundant sub-systems.108 The new sub-systems 235,

532, 523 and 325 are seldom or never used in this music, as the only configuration for

hypersystem 235, with its two systems 235 and 253, seems to be the one which places the largest

interval in the middle ( i.e., sub-systems 253 and 352). If we were to add this criterion ( i.e., if

we dismiss sub-systems containing suites of very small intervals such as 23 or 32, in the quarte-tone model) to the filters already used for the semi-tone model of the fourth, we would end up

having NI=3 as the unique possibility for this containing interval, because the remaining sub-


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systems for NI=4 are excluded by this criterion (see the row intersecting criteria 2 for thequarter-tone model in figure 23, page 188).

The reverse pycnon ruleAll the filters and criteria used with the fourth correspond to common practice and theory

and their application provides with complementary information on the aesthetics of modal music,

especially with the maqmand modal diatonic music. It would be interesting however to try tofind one single criterion which would have the same effect as the four criteria explained

above.109 Taking a closer look at the composition of the sub-systems commonly in use in the

diatonic and Arabian music (fig. 23: 188: quarter-tone model, in bold), and comparing the sums

for any two conjunct intervals within them, we come up with a very interesting conclusion. All

these sums are comprised between 6 and 8 quarter-tones. Fig. 26: 191 shows pairs of conjunct

intervals in ascending values from the top and the left, beginning with a first interval of 2, and

conjunct intervals (from top to bottom), beginning also with a 2, incremented until the maximum

which is 8 quarter-tones, in order to complete the sum for the fourth.

163

The next column shows the same process, starting with interval 3 and a conjunct interval

2, with the conjunct interval incremented by one unit downwards. The largest interval for this

column is 7, since the sum of the two intervals may not exceed 10, which is the value of the

fourth in multiples of the quarter-tone.

The process continues for the other columns until all possibilities are given. Common bi-

interval combinations are written in bold on grey background for combinations commonly used

in Arabian music, or on black background for diatonic genera . Sums are given on the top right

or bottom left corners of each bi-interval element. Equality of the sums follows oblique parallellines, from bottom left to top right (or reciprocally).

All series with two conjunct intervals found in the commonly used tetrachords are

concentrated in the three oblique rows with sums of 6, 7 or 8. Other combinations have sum

values below or above. This is a very strong indicator for homogeneous interval distribution of

the intervals within the scale. If we add to all these bi-interval combinations other intervals, to

the left or to the right and check those which follow the rules of homogeneity (fig. 27: 191), we

end up having only common tetrachords listed in figure 23, page 188. With a single criterion

applied to the intervals of the sub-systems within the fourth, there is a model which is the closest

possible to common practice and theory. Furthermore, this rule of homogeneity is the reciprocal

of Aristoxenus pycnon rule whichsays that a pycnon (a bi-interval scale element composed of

two small intervals within a fourth) must be smaller or equal to the one tone interval.110

The ruleof homogeneity observed with common genera, which we could also qualify as reverse pycnon,

says the contrary (fig. 28: 192). The complement (here of any bi-interval combination inside the

fourth) must have the same limitations as those for Aristoxenus pycnon, and the bi-intervalcombination, although equal to, or greater than the one-tone interval (not a pycnon in

Aristoxenus), has the same limitation as for the complement of the pycnon with Aristoxenus.

This important difference may have one of three causes.

1) With our modern music as with traditional forms, such as with the maqm, there hasbeen important evolution diverging from their initial form, which initially, might have been close

to Aristoxenus descriptions.2) Arabian music and Ancient Greek music were never connected, and the former

evolved independently from the latter, or3) Aristoxenus description of the music of his time was not accurate or had no relation

with practice.111


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Applying the reverse pycnon rule to the fifth and the octave The last filter has shown the

most commonly used genera. Consequently, it would be interesting to apply this principle to the

fifth, or to the octave. This is simple enough, with the fifth, and consists in adding an interval at

one end and at the other of the genus, within the rules of homogeneity (fig. 29: 192), and then

verifying the sums of the resulting combinations.112

As expected, this shows that most of the pentachords resulting form this operation have

their equivalence in both literature and practice, although some of possible pentachords do notappear in the series.113 Figure 30, page 193, shows an example of scale building beginning with

the ijz tetrachord.This is very similar to the generation of fifths, although less than half of the

combinations (with a black background on the figure) exist in the literature or in the practice,

with the remaining scale elements not found in the literature.114 Possibilities for some limited

hexatonic elements (for example 626262 and 262626 in the figure) also exist. As a consequence,

whenever the rule of homogeneity applies to commonly used genera, its extension to the fifth

and octave intervals is either inadequate or too restrictive, although it shows that the full

potential of Arabian music, even with such a restrictive criterion, is still not fulfilled.

There is a noticeable exception with the maqm mukhlif which in Arabic means

infringer, which has a limited scale of b- 3 c 2 db 4 eb 2 f b where the two first intervalsbreach the rule of homogeneity. There are other maqmwhere conjunct tetrachords may formneighbouring semi-tones as for example in the maqm Naw-Athar where the interval/tetrachorddistribution is [4] {262}{262} (or a disjunctive one tone interval followed by two ijztetrachords, where the two neighbouring semi-tones (in italics) also breach this rule (when

applied to the octave). This is the main reason why, although the homogeneity rule is a perfect

match-maker for tetrachords, we shall keep, for the following statistical studies in the frame of a

fifth or an octave, the initial criteria given in figure 23, page 188.115

A little incursion in the eighth-of-a-tone model

The reader may be wondering why this study does not give more refined models, such asthe eighth of a tone, for example. A first answer was given above and said that the purpose of

interval generation was to use the least possible divisions in a containing interval with the utmost

number of combinations, according to the principle of economy.

A second answer comes from the definition of the conceptual interval. Any interval in

use in a scale should be relatively easily identified, both by performers and listeners alike (this

procedure becomes difficult whenever the elementary intervals are smaller than the one-quarter

tone). However, and as a confirmation of the principle of relative size of intervals within a

containing interval, we shall have a quick look at this possibility, in fourth.

164

When dealing with a new interval model, it must be first determined which are

conceptual, elementary, or measurement intervals. When the measurement interval is one-eighth-

of-tone, what would be the smallest conceptual interval? Two-eighths-of-a-tone would be too

small because it equals a quarter-tone which is too small for being conceptual. A three-eighths-

of-a-tone interval, as used by Aristoxenus in his hemiolic chromatic genus,116 with two conjunct

intervals of three-eighths of a tone and one interval of fourteen-eighths or seven-quarters-of-atone, would restrict us to the 17-ET inspired by Urmaws theory (fig. 31: 194, central one toneinterval), with a three-eighths interval equivalent to a limma, an elementary quarter-tone used as

an auxiliary interval, and with two possibilities for the mujannab interval (see the three one-tone

intervals to the right of figure 31, page 194).


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with or without the filters in resulting sub-systems. These filters reduce possibilities and give a

hold on the internal mechanisms of modal music. Interval combinations chosen throughout

history can be described and recognisedtheir positioning and qualitative sizes within the fourthor the fifth is not a coincidence. Furthermore, as we try to reduce the steps between intermediary

intervals (as in the eighth-of--a-tone model of the fourth), the tendency towards a balance of the

generations around the (same) optimal value remains, with however

165

quantitative differences between models. The semi-tone model seems to be best suited to

the fifth, rather than to the fourth: the optimal value in the semi-tonal modelling of the fifth is

very stable and the angle formed by the two bordering segments of the line is acute and (fig. 33:

195) accented in the case of a limited alphabet (fig. 34: 196); this optimal value still exists for the

fourth, in the semi-tone model, but with a very limited number of combinations: in this case,

only four major (diatonic) combinations may be used by the performer, which is somewhat

limited compared to the twelve combinations in the quarter-tone model of the fourth (fig. 23:

188: nine genera are left if we filter the sub-systems with the second set of intersecting criteria).Twelve (or nine) combinations within a fourth seems a suitable reservoir for modal possibilities,

alone or in combination, in performance or as paradigms for a repertoire as it gives the performer

good possibilities for modulations, with the fourth as a starting containing interval that he can

elaborate further and further (by modifying its internal structure or interval composition), andthen perhaps expand the span of the melody to the fifth or more. An eighth-of-the-tone model

gives too many intermediate possibilities while adding perception difficulties (for example, an

eighth-of-a-tone is much more difficult to recognize than a interval of one-quarter-tone, and the

difference between a three-quarter-tone interval and a one-tone interval is much easier to

distinguish from the difference between a six-eighths-of-a-tone interval and a seven-eighths-of-

a-tone interval). With the fifth, however, the quarter-tone model becomes too rich,121 and too

complicated. Almost seventy possible combinations are available to the performer, which wouldbe difficult to memorise.

122It is easier to add a one-tone interval below or above the bordering

intervals of the fourth. (fig. 29: 191). This is a process that would give some fifteen interval

patterns available within the fifth. This is a practical means for enriching the repertoire with the

least possible number of conceptual intervals. Even then, the semi-tone possibilities of the fifth

compete with the potential of this last model. This is mainly due to the fact that the addition of

one tone to the fourth reinforces the diatonic nature of interval combinations, as well as the

possibilities for bi-fourth configurations (two intersecting fourths with successive ranksfig. 35:196).

Should we start our scale element with a one-tone interval (fig. 35: 196, left, the one-tone

interval equates to the 4 quarter-tones), possible combinations complying with both rules of sum

(for the adjacent fourthin order to obtain a fifth) and the rule of homogeneity are more or lessbalanced between elements with neutral intervals (five) and elements with (exclusively) diatonic

intervals (four). If we begin our element (ascending from left to right) with a neutral interval

such as the three-quarter-tone interval (fig. 35: 196, right), the remaining three intervals cannot

make a fourth (their sum is always equal to 11 quarter-tones). In order to make a fifth, we are in

some cases, for example 3345, 3434 and 3524, compelled to complete first the just fourth, then

to add to it the one-tone interval, at the end. This process leaves us with only three possible

combinations having both fourth and fifth, which is very little when compared to the nine

possible combinations in the preceeding case in which we have set the first interval to 4, and in

which all combination have both fourth and fifth.

In an open process, however, not taking into account the fifth as a necessary step on theway to the octave, the reduced potential of the starting neutral three-quar-ter-tone interval widens

up very quickly (before being re-stricted once again by the octave).


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As a preliminary conclusion, we may say that the quarter-tone model is particularly

suited to the just fourth, whilst the semi-tone model is better suited to the fifth as a containing

interval. Both models, however, show that the number of four or three intervals within a fifth or

a fourth, is not coincidental, but it is the result of an optimization process between complexity

and expressivity.

A further remark can be made concerning octave systems of scales. What is applicable to

the fourth also applies to a combination of two fourths with a one tone interval, or tocombinations of fourths and fifths within the octave. Adding up the numbers of optimised

interval repartitions for two fourths (twice three optimal conceptual intervals) + a one-tone

interval, the optimal number of intervals for the octave is seven the same applies to the totaloptimised number of intervals from the combinations of fourths (three optimal intervals) and

fifth

Documents

Amine Beyhom S.urmavi s.151